aft ce gasflow reprint
TRANSCRIPT
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January2000
www.che.com
Trey Walters, P.E.
Applied Flow Technology
APublicationof Chemical WeekAssociates
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Cover Story
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Cover Story
GGaass--ffllooww CCaallccuullaattiioonnss::Trey Walters, P.E.
Applied Flow Technology
Assuming incompressible flow simplifies the math,
but introduces error. Always know how much
when compressibility can be ignored.
Compressible flow: Equation (1) is
not strictly applicable to compress-ible flow because, as already noted,
the density and velocity change along
low of gases in pipe systems is
commonplace in chemical-pro-
cess plants. Unfortunately, the
design and analysis of gas-flowsystems are considerably more
complicated than for liquid (incom-
pressible) flow, due mainly to pressure-induced variations in the gas-stream
density and velocity. Here, we review
practical principles and present some
key equations governing gas flow, and
assess several assumptions and rules
of thumb that engineers sometimes
apply in order to simplify gas-flow
analysis and calculations.
Compressible, incompressibleIn a broad sense, the appropriate term
for gas flow is compressible flow. In a
stricter sense, however, such flow can
be categorized as either incompress-ible or compressible, depending on the
amount ofpressure change the gas un-
dergoes, as well as on other conditions.Accurately calculating truly com-
pressible flow in pipe systems, espe-
cially in branching networks, is a for-midable task. Accordingly, engineers
often apply rules of thumb to a given
design situation involving gas flow, todecide whether the use of (simpler) in-
compressible-flow calculations can be
justified. Such rules of thumb are help-
ful, but they can lead one astray when
used without a full understanding ofthe
underlying assumptions.
Sometimes, the case is clear-cut. For
instance, if the engineer is designing
a near-atmospheric-pressure venti-
lation system, with pressure drops
measured in inches of water, incom-
pressible-flow methods are perfectly
suitable. Conversely, for design or
specification of a pressure-relief sys-
tem that is certain to experience high
velocities, compressible-flow methods
will clearly be required. In practice,
many gas systems fall between these
extremes, and it is difficult to assessthe error that will result from using
incompressible methods.
A major purpose of this article is to
offer guidelines for assessing the im-portance of compressibility effects in a
given case. First, however, we set out
relevant equations, and discuss some
key aspects of gas-flow behavior.1
The underlying equationsIncompressible flow: An apt start-ing point for discussing gas flow is
an equation more usually applied toliquids, the Darcy-Weisbach equation
(see Nomenclaturebox, next page):
(1)
where f is the Moody friction factor,
generally a function of Reynolds num-
ber and pipe roughness. This equation
assumes that the density,, is
constant. The density of a liquid is
a very weak function of pressure
(hence the substance is virtually
incompressible), and density changesdue to pressure are ignored in
practice. The density varies more
significantly with tem- perature. In
systems involving heat transfer, the
density can be based on the
arithmetic average, or, better, the log
mean temperature. When the ap-
propriate density is used, Equation (1)
can be used on a large majority of liq-uid pipe-flow systems, and for gas flow
1. The quantitative compressible- and incom-pressible-flow results in this article were obtainedusing, respectively, AFT Arrow and AFT Fathom.Both are commercially available software forpipesystem modeling. A simplified but highly usefulutility program, Compressible Flow Estimator(CFE), was developed specifically for this article,and was used in several cases.
the pipe. Sometimes, engineers apply
Equation (1) to gas flow by taking the
average density and velocity. But, be-
cause the variation of each of theseparameters along a pipe is nonlinear,
the arithmetic averages will be incor-
rect. The difficult question How
seriously incorrect? is discussed in
detail later in this article.
Individual length of pipe: More strictly
applicable than Equation 1 to gas flow
in a pipe are Equations (2)(6) [13],
developed from fundamental fluid-
flow principles and generalized from
perfect gas equations [4] to apply to
real gases:
Mass:
(2)
Momentum:
(3)
Energy:
(4)
Equation of State:
(5)
Mach number:
(6)
Several things should be noted
about Equations (2)(6):
They assume that the pipe diameter
is constant
They are applicable not only to indi-
vidual gases but also to mixtures, so
long as appropriate mixture proper-
ties are used
Equation (1) is a special case of the
momentum equation, Equation (3). Ifthe third term on the left-hand side
of the latter (commonly called the ac-
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F
NomENClATurE Flow chokes at exit into
A cross-sectional flow area of apipe
D diameter of a pipe
e pipe wall roughness
f friction factor
s entropy
T temperature, static
T0 temperature, stagnation
V velocity Flow chokes at
atmosphere or tank
Ff, g, , T0parameters in Equation (14) arithmetical average of F over
computing section
g acceleration (usually gravita-
tional)
h enthalpy,statich0 enthalpy, stagnation
L length of a pipe
x length
z elevation
Z compressibility factor
specific heat ratio
anglefrom horizontal density
SubSCriPTS
expansion in pipe area
Flow chokes atresriction in pipe
M Mach number
m mass flowrate
P pressure
P0 pressure, stagnation
R gas constant
celeration term) is neglected, the two
equationsbecome identical
Equation (4), the energy equation,includes the conventional thermody-
namic enthalpy plus a velocity term
that represents changes in kinetic
energy. The sum of these two terms is
known as the stagnation enthalpy (see
discussion of stagnation properties,
below). The thermodynamic enthalpy
is referred to as the static enthalpy
(even if it pertains to a moving fluid).
Similarly, temperature in a non-stag-
nation context is referred to as static
temperature
Equation (5), as shown, includes acompressibility factor to correct the
ideal gas equation for real-gasbehavior.In general, however, the real-gasprop-
erties can instead be obtained from a
thermophysical property database
Piping networks: In situations in-
volving a gas-pipe network, Equations
(2)(6) are applied to each individual
pipe, and boundary conditions be-
tween the pipes are matched so that
mass and energy are balanced. The fol-lowing equations describe this balance
at any branch connection:Mass balance:
(7)
Energy balance:
(8)
In Equation (8) (in essence, a state-
mentoftheFirstLaw ofThermodynam-
1 Location 1 inpipe
2 Location 2 inpipe
i junction at which solution issought
j junctions with pipes connecting
to junction i
nection are the same.
If gas streams of different composi-
tion mix at abranch connection, abal-ance equation will also be needed for
each individual species present. Ad-ditional discussion of species balance
can be found in Reference [3]. Use of
these network-calculation principles
is discussed in more detail later.
Besides the use of the basic equa-
tions set out above, gas-flow designs
and calculations also frequently in-
volve two concepts that are usually of
lesser or no importance with incom-
pressible flow: stagnation conditions,
and sonic choking.
Stagnation conditionsAt any point in a pipe, a flowing gas
has a particular temperature, pres-
sure and enthalpy. If the velocity of
the gas at that point were instanta-neously brought to zero, those three
properties would take on new values,
known as their stagnation values and
indicated in the equations of this ar-
ticle by the subscript 0.
Three important stagnation condi-
tions can be calculated, for real as wellas ideal gases, from the velocity and
the specific heat ratio (ratio of specific
heat at constant pressure to that at
constant volume) by Equations (9 a, b
and c). As is frequently the case in gas
flow, the velocity is expressed in terms
of the Mach number:
(9a)
Figure 1. Any of these piping configu-rations can result in sonic choking
Sonic chokingIn almost all instances of gas flow in
pipes, the gas accelerates along the
length of the pipe. This behavior can
be understood from Equations (2), (3)
and (5). In Equation (3), the pressure
falls off, due to friction.As the pressuredrops, the gas density will also drop
(Equation [5]). According to Equation
(2), the dropping density must be bal-anced by an increase in velocity to
maintain massbalance.
It is not surprising, then, that gasflow inpipelines commonly takes place
at velocities far greater than those for
liquid flow indeed, gases often ap-proach sonic velocity, the local speed of
sound. A typical sonic velocity for air
is 1,000 ft/s (305 m/s).
When a flowing gas at some locationin the pipeline experiences a local ve-locity equal to the sonic velocity of the
gas at that temperature, sonic choking
occurs and a shock wave forms. Such
choking can occur in various pipe con-
figurations (Figure 1).
The first case, which can be called
endpoint choking, occurs at the end of
a pipe as it exits into a large vessel or
the atmosphere. In this situation, the
gas pressure cannot drop to match
that at the discharge without the gas
accelerating to sonic velocity. A shockwave forms at the end of the pipe, re-
sulting in apressure discontinuity.
The second case, which might be
called expansion choking, crops up
when the cross-section area of the
pipe is increased rapidly; for example,
if the system expands from a 2-in. pipe
to one of 3-in. pipe. This can also hap-
pen when a pipe enters a flow splitter
such that the sum of the pipe areas on
ics),energyisbalancedbysumming(for
each pipe at the branch connection)the
mass flowrate multiplied by the stag-
nationenthalpy.Elevationeffectsdropout, because all elevations at the con-
(9b)
(9c)
the splitting side exceeds the area of
the supply pipe. A shock wave forms at
the end of the supply pipe, and apres-sure discontinuity is established.
(Continues on next page)
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Cover Story Mach/number
Mach/number
Choked-flow
rate,
lbm
/s
pstag/P
stag
inlet
pstag/Ps
tag
inlet
Choked-flow
rate,
lbm
/s
1 1 1 1
Cover Story 0.8 0.8 0.8 0.8
0.6 0.6 0.6 0.6
0.4 0.4 0.4 0.4
The third case, which may be called 0.2 0.2 0.2 0.2restriction choking, occurs when the
as flows throu h a restriction in the 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1i e, such as an orifice or valve. In x/L x/L
such a case, the flow area of the gasis reduced, causing a local increase in
velocity, which may reach the sonic
Figure 2. These stagnation-pressure and Mach-number profiles are for (left) ex-pansion choking, involving a 2-in. pipe expanding to 3 in., and (right) restriction chok-ing at a 0.6-area-ratio orifice in a 2-in. pipe
velocity. A shock wave forms at the re-striction, with a pressure discontinu-
ity similar to the first two cases.
Figure 2 shows stagnation-pressure
and Mach-number profiles for expan-
sion choking and restriction choking;
both involve supply air at 100 psia
and 1,000R discharging to 30 psia.
Endpoint-choking behavior appears in
Figure 7, discussed later.
For a given process situation, thechoked flowrate can be determined
from Equation (10a), by inserting a
Mach number of 1 into Equation (10b):
(10a)
where:
(10b)
These equations can be derived from
the continuity equation [4,p.97].In practice, it is difficult to apply
these equations to choked conditions,
because the local conditions, P0 and
T0, are not known at the point of chok-ing. For instance, to apply the equa-tions to endpoint choking, one must
calculate the stagnation pressure and
temperature at the end of the pipe, up-stream of the shock wave but these
two variables depend on the flowrate,
which is not yet known.
The only way to solve such a prob-
lem accurately is by trial and error:first, assume a flowrate and march
down the pipe; if M reaches 1 before
the end of the pipe, repeat the proce-dure with a lower assumed flowrate;
repeat until M reaches 1 right at the
pipe endpoint. Obviously, this calcula-
tion sequence is not practical without
a computer.
From the standpoint of pipe design
or system operation, sonic choking
sets a limit on the maximum possible
flowrate for a given set of supply con-
ditions. In particular, lowering the
discharge pressure does not raise the
flowrate. Figure 3 illustrates this for
5
4
3
2
1
00 0.2 0.4 0.6 0.8 1
dP stag/P stag inlet
Figure 3. In this adiabatic flow of 100-psia, 70F air, sonic choking occurs at63.6-psia or lower discharge pressure
a 2-in. pipe carrying air that is sup-
plied at 100 psia. Despite containing
no physical restrictions, this system
experiences endpoint choking at any
discharge pressure below 63.6 psia.
Some engineers misapply the con-
cept of sonic choking and conclude
that the sonic flowrate is the maxi-
mum possible through a given system
for all conditions. In fact, however, the
flowrate can be increased by raisingthe supply pressure. Indeed, the in-
creased choked-flowrate presumably
increases linearly with increased sup-
ply pressure (Figure 4).
The pressure drop across the shock
wave in choked flow cannot be calcu-
lated directly.2 The only recourse is
to use the choked flowrate as a new
boundary condition on the pipe down-
stream of the shock wave (assuming
that one is not dealing with endpoint
choking) and to apply Equations (2)
(6) in the remaining pipes. The shock-wave process is not truly isenthalpic,
but (in accordance with Equation [4])
instead entails constant stagnation
enthalpy.
Be aware that a given pipe can choke
at more than one location along its
length. This occurs when the choked
flowrate set by the upstream choke
point is applied to the pipes beyond
the upstream shock wave, and the
gas at this flowrate cannot reach the
end of the pipe without experiencing2.Normal shock tables (perhaps more familiarto aeronautical engineers than to chemical engi-neers) apply only to supersonic flows, and are ofno use for sonic or subsonicpipe flow.
4
3
2
1
00 100 200 300 400 500 600
Supply pr essure, psia
Figure 4. Increasing the supply pres-sure raises the choked flowrate (shownhere for an adiabatic flow of steam)
another shock wave. In fact, there is
no limit to the number of choke points
in a pipe, other than the number of
possible geometric configurations that
permit shock waves. The three mecha-
nisms that cause choking can all occur
in the same pipeline, in any combina-
tion. References [2] and [3] discuss
calculation procedures for multiple-
choking systems.
Single-pipe adiabatic flowBefore presenting compressible-flow
equations that are generally applica-
ble (Equations [13] and [14]), we con-
sider two special cases: adiabatic and
isothermal flow. Both are important
in their own right. Whats more, analy-
sis of the two (see below) leads to the
guidelines that can help the engineer
decide whether compressibility (with
its far more-complex calculations)
must be taken into account in a given
process situation.The thermodynamic process a gas
undergoes in constant-diameter adia-batic flow can be viewed in terms of
entropy and static enthalpy. This
process traces out a curve called the
Fanno line3 (Figure 5). The Fanno line
neglects elevation changes, a safe as-sumption in most gas systems.
According to the Second Law of Ther-
modynamics, the entropy increases as
the gas flows through the pipe. Thus,
depending on the initial state of the
gas (either subsonic or supersonic), the
3. Some authors show the Fanno line as a plot oftemperature (rather than enthalpy) vs. entropy.
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Case Massflow (lb/s)
Cooled 0.9197Adi abati c 0.8994Isothermal 0.8903Heated 0.8776
T
emperature,
F
h
Sonic point
ho = constant
V2
2
smax s
110
90
70
50
Distance, ft.
Heated IsothermalFigure 6. Adiabaticand isothermal flow do
In four situations shownhere, 100-psia, 111F airis fed into a 1-in. pipe 20 ftlong. Outlet pressure is 60psia. Cooled flow has 30F
ambient temperature;heated flow, 220F. Theheat-transfer coefficients
are 100 Btu/(h)(ft )(F)
Figure 5. Fanno lines, such as the onepresented here, show enthalpy vs. en-tropy for adiabatic flow in a pipe
process will follow either the upper or
lower portion of the curve. Very few
process situations entail supersonic
flow in pipes, so we will focus on the
subsonic (i.e., upper)portion.The stagnation enthalpy, h0, is con-
stant because the system is adiabatic.However, the gas is accelerating, which
causes the staticenthalpy to decrease,
in accordance with Equation (4). If the
proper conditions exist, the gas will
continue to accelerate up to the point
at which its velocity equals the sonic
velocity,where sonic chokingbegins.
As Figure 5 shows, the enthalpy
approaches the sonic point asymp-
totically. Accordingly, the thermody-
namic properties experience intenselyrapid change at the end of a sonically
choked pipe. Examples of such change
arise later in this article.
The gas static temperature usu-ally decreases as it travels along the
pipe, due to the decreasingpressure.
Under certain conditions, however, the
(11b)
Single-pipe isothermal flowIn the second special case, isothermal
flow, the static temperature of the gas
remains constant. As already noted,
the tendency is for gas to cool as it
flows along a pipe. For the tempera-
ture to remain constant, an inflow of
heat is required.
When temperature is constant,
Equations (2)(6) become somewhat
simpler. In Equation (5), for instance,
density becomes directly proportional
to pressure, and a perfect-gas analyti-
cal solution can be obtained:
(12a)
where the T subscript on L empha-
sizes that the system is isothermal.
Integrating from 0 to L gives:
consider the adiabatic case, where no
heat is added but the gas cools. If heat
is removed, the cooling will exceed
that in adiabatic flow. Next consider
isothermal flow, where the addition of
heat keeps the gas static temperature
constant. If more heat is added than
required to maintain isothermal flow,the static temperature will increase.
In summary, the heat-transfer envi-ronment plays a critical role in deter-
mining whether the gas flow is closer
to adiabatic or isothermal. It is also
the mechanism that can cause the gas
flow to exceed the limits of the two
special cases. Figure 6 demonstrates
the different situations.
General single-pipe equationsFor the general (neither adiabatic nor
isothermal) case, in situations whenthe compressibility of the gas can-not be ignored, Equations (2)(6) can
be combined and, through calculus
and algebra [3, 4], represented in dif-ferential form by Equations (13) and
(14). Equation (13a) [1-3] is based on
a fixed-length step between Locations
reverseistrue.Thegoverningparam- 1 and 2 along the pipe. The terms in-
eterinthisregardistheJoule-Thomp-
son coefficient [5, 8]. The points made
in this article are (unless otherwise
(12b) volving and Z account for the
real- gas effects:
noted) applicable for either the cool-
ing or heating case if the appropriatewords are substituted, but we assume
the cooling case for the sake of discus-sion. For more on Fanno flow see Ref-
erences [4, 6, 7].
From Equations (2) (6), the follow-ing equation can be derived for adia-
batic flow of a perfect gas [4, p. 209]:
To truly maintain isothermal flow
up to the sonic point would require aninfinite amount of heat addition. This
leads to the strange but mathemati-cally correct conclusion that for iso-
thermal flow, sonic choking occurs at a
Mach number less than 1. Practically
speaking, it is not feasible to keep a
gas flow fully isothermal at high veloc-
ities. For a more-complete discussion
Integration yields:
where:
(13a)
(13b)
(11a)
Integrating from 0 to L along the
ofisothermal flow inpipes,seeRefer-
ence [4], pp. 265269.
One occasionally finds a misconcep-tion among engineers designing gas
systems: that adiabatic and isother-
mal flow bracket all possible flow-
(13c)
length of the pipe gives: rates. However, this is not true. First, Conditions at Location 1 are known;
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Incompressible-flow
rateerror,%
Flow
rateerror,%
Error,%
30
25L/D = 50 Air
400.1 0.5
1.0
30
20 T
/T = 0.6
L/D = 20020
15
10
5
0
SteamMethane
L/D = 1000
30Sonic choking above this line
1.5
20fL/D=3
5
10 1015
30
050
10
0
-10
-20
-30
ambient inlet
0.8
1.2
1.5
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6
dPstag/Pstag, inlet dPstag/Pstag, inlet dPstag/Pstag, inlet
FIGURE 8. The pipe pressure-drop ratio
and the ratio of pipe length to diameterare appropriate parameters for general-izing about the error that is introducedwhen assuming incompressible flow
FIGURE 9. Thismap shows the error
(overprediction) in flowrate predictionfora single pipe due to using incompress-ible-flow assumptions rather than anadiabatic compressible-flowcalculation
Figure10. When a pipe istreated as adia-
batic but actually has heat transfer,the flow-rate predictionerror can be sizable, even with-out an incompressible-flowassumption. Thecase here isfor 100-psia,70F air enteringanuninsulatedsteel pipe with L/D ratio of 200
incompressible-flow-assumption error
against this parameter makes itpossi-
ble to summarize the information on a
single curve for each fL/D value, which
applies for all pipe diameters.
Such an error map appears in Fig-
ure 9. It is based on an iterative pro-gram, Compressible Flow Estimator
(CFE), developed by the author andbeing made available as a free down-load athttp://www.aft.com/cfe.htm.
The results shown in Figure 9 are
of general applicability. Various spe-
cific heat ratios, , and
compressibility factors, Z, have been
entered into the CFE, and the results
always fall along the lines shown in
Figure 9. This error map is also
consistent with real-sys- tem
predictions based on more-sophis-
ticated calculation methods. Accord-
ingly, Figure 9 is recommended to theengineer for general use as a guide in
assessing compressibility in pipes.
Keep in mind, though, that Figure
9 assumes adiabatic flow. Additional
error can result from flows involving
heat transfer. The relative importance
of heat transfer is addressed in the
next section.
Finally, note that the direction of
the incompressible-flow-assumption
error is to overpredict the flowrate. Or,
stated differently, for a given flowrate,
it will underpredict the pressure drop.Unfortunately for typical pipe-system
applications, neither of these conclu-
sions is consistent with conservative
design.
The sequence of steps that underlie
the CFE program are available from
the author.Also available from him are
modified sequences, for handling situ-
ations in which (1) the endpoint static
pressure rather than the stagnationpressure are known, or (2) the temper-
ature and flowrate are known but the
endpoint stagnationpressure is not.
Effectof heat transfer:The author
knows of no general relationship
showing the effect of heat transfer on
the size of the incompressible-flow-as-
sumption error. However, some insight
can be gained from comparing rel-
evant compressible-flow calculations
(setting aside for a moment our pre-occupation with the incompressible-
flow-assumption error). Computer
models were constructed to determine
the difference in flowrate for air at dif-ferent ambient temperatures.
The difference in flowrate for air
with different ambient temperatures
as compared to the compressible adia-
batic case appears in Figure 10. It can
be seen that cooling a gas may resultin a greatly increased flowrate. In
contrast, heating a gas can cause the
flowrate to decrease significantly.
Accordingly, if an engineer is trying
to design for a minimum flowrate, a gas
stream that is cooling works in his or
her favor by causing an underpredic-tion of the flowrate when using adia-
batic flow methods. When this erroris
combined with that of an incompress-
ible-flow assumption, which overpre-dicts the flow, these twoerrorswork in
opposite directions, in part cancelling
each other out. Conversely, a gas being
heated adds furthererroron top of the
incompressible-flow-assumption error,
causing even more overprediction of
the flowrate.
In many gas-pipe-system designs,
the delivery temperature is as impor-tant as the delivery flowrate and pres-
sure. In those cases, the heat-transfer
characteristics of the pipe system take
on the highest importance, and nei-
ther adiabatic nor isothermal methods
let alone incompressible-flow as-
sumptions can give accuratepredic-
tions. Unless the gas flow is very low
and can be adequately calculated with
incompressible methods, the designer
is left with no choice but to perform
a full compressible flow calculation.
This means solving Equations (2)(6)
with a suitable relationship for the
heat transfer to be used in Equation
(4), or using more-convenient forms
of these equations, such as Equations
(13) or (14). Realistically, this requires
appropriate software.
Network complicationsWhen applying the concepts in this ar-
ticle, and in particular the use of the
CFE program that underlies Figure9, to a pipe network, the number of
variables increases and the difficulty
in assessing the potential error like-
wise increases. To investigate possible
error-estimating methods, we have
constructed simple flow models, one
for incompressible flow and the other
for compressible flow, of a manifolding
pipe system. For simplicity, the com-pressible-flow model assumed that all
flows are adiabatic. The basis is a 110-
psia air system that enters a header
and flows to three pipes at successivepoints along the header, terminating
in a known pressure of 90psia.
For each pipe in the system, the
predicted fL/D and pressure-drop
ratio have been determined from the
incompressible-flow model. The re-
sulting data have been entered intothe CFE program for each pipe, and
an approximate error generated for
each. Then, starting from the supply,
a path has been traced to each termi-
nating boundary (of which there are
three). The error for each pipe in thepath has been summed, and then di-
vided by the number of pipes in the
path, giving an average error. This
average has been compared to the
actual difference between the results
of the incompressible- and compress-
ible-flow models.
Overall the comparison has proved
favorable. However, applying CFE to
this networked system underpredicts
the actualerrorfrom the detailed mod-
els by up to 20%. The first pipe in the
header shows the largest error,and the
last pipe the smallest. As in the single-
pipe calculations, the incompressible
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Cover Story
method overpredicts the flowrate.
In short, extra care should be taken
when interpreting the meaning of in-
compressible-flow methods applied to
gas pipe networks.
Rethinking the rules of thumbThe information presented up to now
provides abasis for critiquing a num-ber of rules of thumb upon which en-gineers often depend when dealing
with gas flow.
Adiabatic and isothermal flow:
One rule of thumb is the myth that
adiabatic and isothermal flowbracket
all flowrates. They do not, as has al-
readybeen noted.
40%-pressure-drop rule: A commonbelief is what can be called the 40%-
pressure-drop rule. Presented in a va-riety of handbooks, it states that if the
pipe pressure drop in a compressible-
flow system is less than 40% of the
inlet pressure, then incompressible-
flow calculation methods can be safely
employed, with the average density
along the pipe used in the equations.
In the handbooks, it is not made
clear whether the pressure drop ratio
is to be based on the stagnation or the
static pressures. (In the authors expe-rience, engineers apply the rule more
frequently using stagnation-pressure
ratios.) In any case, Figures 8 and 9
make it clear that the 40%-pressure-
drop rule has no validity unless as-
sociated with a specific L/D ratio. Ac-
cordingly, this rule of thumb is highly
misleading, and should be discarded
by the engineering community.
Choked air flow at 50% pressure
drop: An equation sometimes used
as a rule of thumb to assess the likeli-
hood of sonic choking is as follows (see,for instance, Reference [4], p 94):
(15)
where p* is the critical staticpressure
at sonic velocity and p0 the local stag-nation pressure. For air, the specific
heat ratiois 1.4, so the pressure ratioin
the equation works out to 0.5283. This
However, Equation (15) breaks
down for pipe-system analysis when
pipe friction becomes a factor. The
reason is that the stagnation pressure
in the equation is the pressure at theupstream side of the shock wave. If
there is any pressure drop in the pipe
from the supply pressure to the shock
wave, then the supply pressure cannot
be used in Equation (15). Instead, the
local stagnation pressure at the shock
wave must be used but this is not
known, unless the pressure drop is
calculated using other means.
In short, Equation (15) cannot be
used to predict the supply and dis-
charge pressures necessary for sonic
choking unless the piping has negli-gible friction loss.
Other simplified compressible-flow
methods: A variety of simplified gas-flow equations, often based on assuming
isothermal flow, crop up in thepractical
engineering literature. These typicallyhave several drawbacks that are not al-
ways acknowledged or recognized:
Most gas flows are not isothermal.
In such cases, one cannot know how
much error is introduced by the as-sumption of constant temperature.
Related to this is the general issue ofthe importance of heat transfer on the
gas flow, already mentioned
Simplified equations typically do
not address sonic-choking issues
These equations are of no help when
the delivery temperature is important
The simplified equations break
down at high Mach numbers
Unrealistically, the entire pipe is
solved in one lumped calculation,
rather than using a marching solu-
tion
It is difficult to extend the equationsto pipe networks
In summary, simplified compressible-
flow equations can be an improvement
over assuming incompressible flow,
but numerous drawbacks limit their
usefulness.
Final thoughtsCompressors, blowers and fans raise
ible-flow methods and estimation equa-
tions in this article to such systems.
The methods discussed in this article
can help the engineer assess endpoint
sonic choking, but restriction and ex-pansion choking are somewhat more
complicated. Accordingly, the estima-tion methods in this article may notbe
applied to all choking situations.
For new designs that require a lot of
pipe, the engineer should consider the
potential costs savings if smaller pipe
sizes can be used. If significant cost
savingsprove to be possible, it may be
prudent to invest in developing a de-tailed model that can more accurately
determine the system capability over
a range of pipe sizes. A detailed modelmay also help assess the wisdom of
making modifications proposed for an
existing system.
Edited by Nicholas P. Chopey
References:1. Winters, B.A., and Walters, T.W., X-34 High
Pressure Nitrogen Reaction Control SystemDesign and Analysis, NASA Thermal FluidAnalysis Workshop, Houston, Tex., 1997.
2. Walters, T.W., and Olsen, J.A., ModelingHighly Compressible Flows in Pipe Net-works Using a Graphical User Interface,ASME International Joint Power GenerationConference, Denver, Colo., 1997.
3. AFT Arrow 2.0 Users Guide, Applied FlowTechnology, Woodland Park, Colo., 1999.
4. Saad, M.A., Compressible Fluid Flow, 2ndEd., Prentice-Hall, Englewood Cliffs,NJ, 1993.
5. Carroll, J.J., Working with Fluids that WarmUpon Expansion, Chem. Eng., pp. 108114,September 1999.
6. Anderson, J.D., Jr., Modern CompressibleFlow: With Historical Perspective,McGraw-Hill, New York, N.Y., 1982.
7. Shapiro, A.H., The Dynamics and Thermo-dynamics of Compressible Fluid Flow, 2vols., Ronald,New York, N.Y., 1953.
8. Barry, John, Calculate Physical Properties forReal Gases, Chem. Eng., pp. 110114, April1999.
9. Flow of Fluids Through Valves, Fittings, andPipe, Technical Paper No. 410, Crane Co., Jo-
liet, Ill., 1988.
AuthorTrey Walters, P.E., is Presi-dent and Director of SoftwareDevelopment for AppliedFlow Technology (AFT, 400W. Hwy 24, Suite 201, P.O.Box 6358, Woodland Park,CO 80866-6358; Phone: 719-686-1000; Fax: 719-686-1001;E-mail: [email protected]).He founded the company, adeveloper of Microsoft-Win-
results in a pressure drop ratioof near the system pressure and density.These lation software,dows-based pipe-flow-simu-
in 1993. Previously, he was a
47% (in otherwords,about 50%) tobringabout sonic choking. For gases with dif-
ferent specific heat ratios, the pressure
drop ratio will differ somewhat, in ac-
cordance with Equation (15).
changes in properties inside the gas-flow system furtherlimit the applicabil-
ity of incompressible methods, beyond
the cautions already discussed. Take
special care in applying the incompress-
senior engineer in cryogenic rocket design forGeneral Dynamics, and a research engineer insteam-equipment design for Babcock & Wilcox.Walters holds B.S. and M.S. degrees in mechani-cal engineering from the University of Californiaat Santa Barbara, and is a registered engineerin California.
Reprinted from the January 2000 issue of Chemical Engineering 2006 Access Intelligence LLC
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